Alan Weinstein (UC Berkeley and Stanford), April 6, 1:00 pm EDT

From symmetry to constraints in general relativity

Abstract: In general relativity, the gravitational field is a Lorentz metric on a 4-dimensional space-time manifold. The Einstein field equations may be expressed, at least locally in time, as the trajectories of a hamiltonian system on the cotangent bundle T∗R of the (infinite dimensional) manifold R of riemannian metrics on a 3-dimensional “time slice”, with initial values constrained to a (singular) submanifold C of T∗R.

Geometric properties of C suggest that the constraints should be related to the symmetry group of the Einstein equations, consisting of the diffeomorphisms of space-time. But this group does not act on R, since it does not act on an individual time slice. Blohmann, Fernandes, and I showed that the algebraic structure of the constraints is in fact related to a groupoid of diffeomorphisms between pairs of time slices, but a direct connection between the constraints and this groupoid was not found.

In a colloquium talk here two years ago, I reported on work on Hamiltonian Lie algebroids which seemed to establish the missing connection, but unfortunately a closer look showed that the application of our general theory to the relativity problem was not quite correct. In this talk, I will describe ongoing work with Christian Blohmann (Bonn) and Michele Schiavina (Zurich) in which we are pursuing a related approach which looks more promising.

The Hamiltonian Lie algebroid which we have constructed lives over the product of T∗R with a space whose coordinates are infinitesimal of first order; i.e., all products among them are zero. This space is a slight modification of a supermanifold which arises naturally via the BFV (Batalin-Fradkin-Vilkovisky) theory of bound- ary value problems for field theories with symmetries which do not preserve the boundary.